Announcements. Image Formation: Outline. Earliest Surviving Photograph. Compare to Paintings. How Cameras Produce Images. Image Formation and Cameras

Transcription

1 Announcements Imge Formtion nd Cmers CSE 252A Lecture 3 Pizz Course reserves vilble Instructor office hours TBD Homework 0 is due tody by :59 PM Wit list Red: Chpters & 2 of Forsyth & Ponce Chpter 2 of Szeliski (Optionl) Imge Formtion: Outline Fctors in producing imges Projection Perspective/Orthogrphic Projection Vnishing points Projective Geometry Rigid Trnsformtion nd SO(3) Lenses Sensors Quntiztion/Resolution Illumintion Reflectnce nd Rdiometry Erliest Surviving Photogrph First photogrph on record, l tble service by Nicephore Niepce in 822. Note: First photogrph by Niepce ws in 86. Compre to Pintings How Cmers Produce Imges Bsic process: photons hit detector the detector becomes chrged the chrge is red out s brightness Willem Klf, Mid 600 s Pedro Cmpos, Sensor types: CCD (chrge-coupled device) high sensitivity high power cnnot be individully ddressed blooming CMOS simple to fbricte (chep) lower sensitivity, lower power cn be individully ddressed

2 Imges re two-dimensionl ptterns of brightness vlues. Effect of Lighting: Monet Figure from US Nvy Mnul of Bsic Optics nd Opticl Instruments, prepred by Bureu of Nvl Personnel. Reprinted by Dover Publictions, Inc., 969. They re formed by the projection of 3D objects. Chnge of Viewpoint: Monet Pinhole Cmer: Perspective projection Abstrct cmer model - box with smll hole in it Hystck t Chilly t sunrise (865) Forsyth&Ponce Cmer Obscur Cmer Obscur "When imges of illuminted objects... penetrte through smll hole into very drk room... you will see [on the opposite wll] these objects in their proper form nd color, reduced in size... in reversed position, owing to the intersection of the rys. --- Leonrdo D Vinci Used to observe eclipses (e.g., Bcon, ) By rtists (e.g., Vermeer). (Russell Nughton) 2

3 Cmer Obscur Distnt objects re smller Jetty t Mrgte Englnd, (Jck nd Beverly Wilgus) (Forsyth & Ponce) Purely Geometric View of Perspective Geometric properties of projection 3-D points mp to points 3-D lines mp to lines Plnes mp to whole imge or hlf-plne Polygons mp to polygons The projection of the point P on the imge plne is given by the point of intersection P of the ry defined by PO with the plne. Importnt point to note: Angles & distnces not preserved, nor re inequlities of ngles & distnces. Degenerte cses: line through focl point project to point plne through focl point projects to line Eqution of Perspective Projection A Digression Crtesin coordintes: We hve, by similr tringles, tht (x, y, z) -> (f x/z, f y/z, f ) Estblishing n imge plne coordinte system t C ligned with i nd j, we get Projective Geometry nd Homogenous Coordintes 3

4 Wht is the intersection of two lines in plne? Do two lines in the plne lwys intersect t point? A Point No, Prllel lines don t meet t point. Cn the perspective imge of two prllel lines meet t point? ES Projective geometry provides n elegnt mens for hndling these different situtions in unified wy, nd homogenous coordintes re wy to represent entities (points & lines) in projective spces. Projective Geometry Axioms of Projective Plne. Every two distinct points define line 2. Every two distinct lines define point (intersect t point) 3. There exists three points, A,B,C such tht C does not lie on the line defined by A nd B. Different thn Eucliden (ffine) geometry Projective plne is bigger thn ffine plne includes line t infinity Projective Plne Affine = Plne + Line t Infinity Homogenous coordintes A wy to represent points in projective spce Use three numbers to represent point on projective plne Why? The projective plne hs to be bigger thn the Crtesin plne. How: Add n extr coordinte e.g., (x,y) -> (x,y,) Impose equivlence reltion (x,y,z) *(x,y,z) such tht ( not 0) i.e., (x,y,) (x, y, ) Point t infinity zero for lst coordinte e.g., (x,y,0) Why do this? Possible to represent points t infinity Where prllel lines intersect Where prllel plnes intersect Possible to write the ction of perspective cmer s mtrix 4

5 Homogenous coordintes A wy to represent points in projective spce Use three numbers to represent point on projective plne Add n extr coordinte e.g., (x,y) -> (x,y,) Impose equivlence reltion (x,y,z) *(x,y,z) such tht ( not 0) i.e., (x,y,) (x, y, ) (x,y,) (x,y) Conversion Eucliden -> Homogenous -> Eucliden In 2-D Eucliden -> Homogenous: (x, y) -> k (x,y,) Homogenous -> Eucliden: (x, y, z) -> (x/z, y/z) In 3-D Eucliden -> Homogenous: (x, y, z) -> k (x,y,z,) Homogenous -> Eucliden: (x, y, z, w) -> (x/w, y/w, z/w) (x,y,) (x,y) Points t infinity Lines in Projective spce Point t infinity zero for lst coordinte (x,y,0) nd equivlence reltion (x,y,0) *(x,y,0) No corresponding Eucliden point Projective Plne Affine = Plne + (x,y,) (x,y,0) Line t Infinity. Line in Eucliden plne 2. Plne through origin in homogenous coordintes 3. Plne is represented by its norml N 4. Eqution for plne is N. (x,y,z) = 0 or M. (x,y,z) = 0 where M = λn N Projective trnsformtion The eqution of projection 3 x 3 liner trnsformtion of homogenous coordintes Points mp to points, lines mp to lines u u2 u x 23 x2 33 x3 Crtesin coordintes: Homogenous Coordintes nd Cmer mtrix U V W 0 0 f 0 T 5

6 Prllel lines meet in the imge Vnishing point End of the Digression Imge plne Formed by line through O Prllel to the given line(s) A single line cn hve vnishing point Vnishing points Vnishing Points H VPL VPR Different directions correspond different vnishing points VP VP 2 VP 3 Vnishing Point In the projective plne, prllel lines meet t point t infinity. The vnishing point is the perspective projection of tht point t infinity, resulting from multipliction by the cmer mtrix. Projective trnsformtion 3 x 3 liner trnsformtion of homogenous coordintes Points mp to points, lines mp to lines u u2 u x 23 x2 33 x3 6

7 Mpping from Plne to Plne under Perspective is given by projective trnsform H Plnr Homogrphy = Hx x =H =H 2 Figure borrowed from Hrtley nd issermn Multiple View Geometry in computer vision x = H 2 = H 2 (H - x) = (H 2 H - )x Figure borrowed from Hrtley nd issermn Multiple View Geometry in computer vision Plnr Homogrphy: Pure Rottion Appliction: Pnorms Figure borrowed from Hrtley nd issermn Multiple View Geometry in computer vision x = H 2 = H 2 (H - x) = (H 2 H - )x 7